Results 1 
6 of
6
Selfdual codes and invariant theory
 MATH. NACHRICHTEN
, 2006
"... There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
There is a beautiful analogy between most of the notions for lattices and codes and it seems to be quite promising to develop coding theory analogues of concepts known in the theory of lattices and modular forms and vice versa. Some of these analogies are presented in this short note that intends to survey recent developments connected to my talk HeckeOperators for codes in Luminy, on May 9, 2007, where I introduce the KneserHeckeOperators mentioned in Section 3.5. More details can be found in the paper [7], a preprint of which is available on my homepage.
KneserHeckeoperators in coding theory
 Abh. Math. Sem. Univ. Hamburg
, 2006
"... The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that inter ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that intersect C in a codimension 1 subspace. The eigenspaces of this selfadjoint linear operator may be described in terms of a codingtheory analogue of the Siegel Φoperator. MSC: 94B05, 11F60
Higher Weights and Binary SelfDual Codes
, 2002
"... The theory of higher weights is applied to binary selfdual codes. Bounds are given for the second minimum higher weight and a Gleason type theorem is derived for the second higher weight enumerator. The second weight enumerator is shown to be unique for the putative [72; 36; 16] Type II code and th ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The theory of higher weights is applied to binary selfdual codes. Bounds are given for the second minimum higher weight and a Gleason type theorem is derived for the second higher weight enumerator. The second weight enumerator is shown to be unique for the putative [72; 36; 16] Type II code and the first three minimum weights are computed for optimal codes of length less than 28. We connect them to the code polynomials of the usual Hamming weight and derive several properties including the MacWilliams relations and the structure theorems of the graded rings associated to the code polynomials of higher weights for small genera, one of which is not CohenMacaulay.
Note on the Biweight Enumerators of SelfDual Codes over Z_k
, 1999
"... Recently there has been interest in selfdual codes over finite rings. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced as a generalization of the biweight enumerators. We investigate these weight enumer ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Recently there has been interest in selfdual codes over finite rings. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced as a generalization of the biweight enumerators. We investigate these weight enumerators and the biweight enumerators of selfdual codes over Z k . The biweight enumerator of a class of binary codes, introduced in this note, is also studied. We derive Gleasontype theorems for the corresponding biweight enumerators with the help of invariant theory. 1 Introduction The conditions satisfied by the biweight enumerators of binary Type I codes and Type II codes were studied in [7] and [5], respectively. Using invariant theory, a basis for the space of invariants, which the biweight enumerator for such codes belongs, was also given. In this note, gfold joint weight enumerators and gfold multiweight enumerators of codes over the ring Z k of integers modulo k are introduced ...
An analogue of Heckeoperators in coding theory.
, 2005
"... The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that inter ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The KneserHeckeoperator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of selfdual codes of fixed length. It maps a linear selfdual code C over a finite field to the formal sum of the equivalence classes of those selfdual codes that intersect C in a codimension 1 subspace. The eigenspaces of this selfadjoint linear operator may be described in terms of a codingtheory analogue of the Siegel Φoperator. MSC: 94B05, 11F60
Selfdual codes and invariant theory 1
"... Abstract. A formal notion of a Typ T of a selfdual linear code over a finite left Rmodule V is introduced which allows to give explicit generators of a finite complex matrix group, the associated CliffordWeil group C(T) ≤ GLV (C), such that the complete weight enumerators of selfdual isotropi ..."
Abstract
 Add to MetaCart
Abstract. A formal notion of a Typ T of a selfdual linear code over a finite left Rmodule V is introduced which allows to give explicit generators of a finite complex matrix group, the associated CliffordWeil group C(T) ≤ GLV (C), such that the complete weight enumerators of selfdual isotropic codes of Type T span the ring of invariants of C(T). This generalizes Gleason’s 1970 theorem to a very wide class of rings and also includes multiple weight enumerators (see Section 2.7), as these are the complete weight enumerators cwem(C) = cwe(R m ⊗ C) of R m×mlinear selfdual codes R m ⊗C ≤ (V m) N of Type T m with associated CliffordWeil group Cm(T) = C(T m). The finite Siegel �operator mapping cwem(C) to cwem−1(C) hence defines a ring epimorphism �m: Inv(Cm(T)) → Inv(Cm−1(T)) between invariant rings of complex matrix groups of different degrees. If R = V is a finite field, then the structure of Cm(T) allows to define a commutative algebra of Cm(T) double cosets, called a Hecke algebra in analogy to the one in the theory of lattices and modular forms. This algebra consists of selfadjoint linear operators on Inv(Cm(T)) commuting with �m. The Heckeeigenspaces yield explicit linear relations among the cwem of selfdual codes C ≤ V N.